We present a CO(1-0) survey for cold molecular gas in a representative sample of 13 high-z radio galaxies (HzRGs) at 1.4<z<2.8, using the Australia Telescope Compact Array. We detect CO(1-0) emission associated with five sources: MRC 0114-211, MRC 0152-209, MRC 0156-252, MRC 1138-262 and MRC 2048-272. The CO(1-0) luminosities are in the range $L_{rm CO} sim (5 - 9) times 10^{10}$ K km/s pc$^{2}$. For MRC 0152-209 and MRC 1138-262 part of the CO(1-0) emission coincides with the radio galaxy, while part is spread on scales of tens of kpc and likely associated with galaxy mergers. The molecular gas mass derived for these two systems is M$_{rm H2} sim 6 times 10^{10}, {rm M}_{odot}$ (M$_{rm H2}$/$L_{rm CO}$=0.8). For the remaining three CO-detected sources, the CO(1-0) emission is located in the halo (~50-kpc) environment. These three HzRGs are among the fainter far-IR emitters in our sample, suggesting that similar reservoirs of cold molecular halo gas may have been missed in earlier studies due to pre-selection of IR-bright sources. In all three cases the CO(1-0) is aligned along the radio axis and found beyond the brightest radio hot-spot, in a region devoid of 4.5$mu$m emission in Spitzer imaging. The CO(1-0) profiles are broad, with velocity widths of ~ 1000 - 3600 km/s. We discuss several possible scenarios to explain these halo reservoirs of CO(1-0). Following these results, we complement our CO(1-0) study with detections of extended CO from the literature and find at marginal statistical significance (95% level) that CO in HzRGs is preferentially aligned towards the radio jet axis. For the eight sources in which we do not detect CO(1-0), we set realistic upper limits of $L_{rm CO} sim 3-4 times 10^{10}$ K km/s pc$^{2}$. Our survey reveals a CO(1-0) detection rate of 38%, allowing us to compare the CO(1-0) content of HzRGs with that of other types of high-z galaxies.
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